Tuning Energy Relaxation along Quantum Hall Channels

1.1 MB·English

by  C. Altimiras

Tuning Energy Relaxation along Quantum Hall Channels
C. Altimiras, H. le Sueur, U. Gennser, A. Cavanna, D. Mailly, and F. Pierre∗
CNRS, Laboratoire de Photonique et de Nanostructures (LPN) - Phynano team, route de Nozay, 91460 Marcoussis, France
(Dated: January 4, 2011)
ThechiraledgechannelsinthequantumHallregimeareconsideredidealballisticquantumchan-
nels,andhavequantuminformationprocessingpotentialities. Here,wedemonstrateexperimentally,
at filling factor νL =2, the efficient tuning of the energy relaxation that limits quantum coherence
andpermitsthereturntowardequilibrium. Energyrelaxationalonganedgechanneliscontrollably
1 enhanced by increasing its transmission toward a floating ohmic contact, in quantitative agree-
1 ment with predictions. Moreover, by forming a closed inner edge channel loop, we freeze energy
0 exchanges in the outer channel. This result also elucidates the inelastic mechanisms at work at
2 νL=2, informing us in particular that those within the outeredge channel are negligible.

n
a PACSnumbers: 73.43.Fj,72.15.Lh,73.23.Ad,73.43.Lp
J

3
Nanocircuits atlow temperature exhibitnew phenom-

] ena resulting from the quantum nature of the transport.
l
l Inmesoscopicphysics,amajorobjectiveistomakeuseof
a
these phenomena to develop novel quantum functionali-
h
tiesforthefuturenanoelectronics. Quantuminformation
-
s processing,whichhas the potential to outperformclassi-
e
m cal computing [1], is a striking illustration. A key task
toreachthisobjectiveistofindefficientwaystoenhance
.
t the robustnessofquantumeffects. It is alsonecessaryto
a
m increase controllably the relaxation rate toward equilib-
rium, e.g. to perform fast resets. Here, we demonstrate
-
d experimentally the up-down controlof energy relaxation
n along a single quantum channel realized in the integer
o
quantum Hall regime (QHR).
c
[ In the QHR, electronsconfined to two dimensions and

immersed in a strong perpendicular magnetic field prop-
2
v agate in chiral channels along the sample edges [2]. The
4 electrical current circulates without dissipation along
7 these edge channels (EC), generally considered as ideal
9
ballistic quantum channels [3]. The similitude between
0 FIG. 1. (Color online) Sample e-beam micrograph: metallic
ECsandlightbeamshasinspirednovelelectronicdevices
. gatesappearbright;thewidegatesontheleftandrightofthe
7 [4–6], and proposals for quantum information processing
0 quantum dot (QD) are grounded and can be ignored. Elec-
0 [7]. However, recent experiments found that quantum tronicexcitationspropagatecounterclockwisealongtwoedge
1 interferences with edge states are more fragile than an- channels (EC), depicted by lines. Dashed lines connecting
: ticipated [4, 6, 8–10]. This observation, corroborated by ECs indicate transmission through quantum point contacts
v
the recent observation of strong energy exchanges be- (QPC).AttheoutputoftheinjectionQPC,theenergydistri-
i
X tween copropagating ECs [11], and possibly with other butionfinj isadoublestep(leftinset)inthehalftransmitted
outerEC.Electronicexcitationstravelalongadjustablepaths
ar thermalizedstates atfilling factorνL =1[12],seemingly from theinjection QPCtotheQD,andpartof theouterEC
impedesthepotentialitiesoftheseECsforquantumelec-
can be diverted toward a floating ohmic contact by tuning
tronics. However, in the present work we show how to the intermediate QPC’s transmission. Right inset: the tun-
tuneefficientlytheenergyrelaxationandtherebythede- nelcurrentIQD throughtheQDisproportionaltotheenergy
phasing,withregardstobothincreasingandfreezingthe distribution fQD probed at theQD in theleft outer EC.
energy exchanges along an EC at ν = 2, where two
L
copropagating ECs are present.
The basic experimental principle to probe energy re- measuring at the path’s end the electronic energy dis-
laxation is sketched in Fig. 1 and detailed in the previ- tribution f (E) in the outer EC with a quantum dot
QD
ous works [11, 13]. It consists in driving the outer EC (QD) operated as an energy filter. At the output of the
out-of-equilibriumwithavoltagebiased‘injection’quan- injectionQPC,whoseconductance is setto 0.5e2/h such
tum point contact(QPC) and,after various propagation that the outer EC is half transmitted and the inner EC
paths defined mostly by voltage biased metal gates, in fully reflected, the electronic energy distributions in the
2
outer and inner ECs are, respectively, a non-equilibrium (a) L=2.2 µm
smeared double step f (see left inset in Fig. 1) and
inj
a cold Fermi function [13]. Energy exchanges along the
edge are revealed through changes in f (E) with the
QD QD
length of the propagation path between injection QPC
and QD. In the actual measurements, we record the
differential conductance ∂I /∂V ∝ ∂f /∂E, with δV
QD p QD
V ∝ E the voltage applied to a plunger gate coupled
p
totheQD,andIQD thetunnelcurrentacrosstheQD.In (b)
the following, we display these raw data normalized by
δV=36 µV
the measured maximum QD current Imax. Full details
QD
regardingthe f spectroscopyaregivenin[11,13]. See
QD
[14]forspecificadditionalinformationonQDcalibration.
The sample shown in Fig. 1 was patterned by e-beam
lithography on a standard GaAs/Ga(Al)As two dimen-
sional electron gas of density 2 1015 m−2 and mobility
250 m2V−1s−1. The same sample was used to demon-
strate non-equilibrium EC spectroscopy [13] and energy
relaxation [11] in the QHR. Conductance measurements
p
were performed using standard low frequency lock-in
techniques in a dilution refrigerator of base temperature
30 mK and near the center of the ν = 2 plateau. To
L
avoid artificial heating, AC voltages were always kept G h/e
i
smaller than k T/e.
B
Energy relaxation with a voltage probe. We
show that we can controllably drive the system up to
full relaxation by diverting the non-equilibrium EC to-
ward a floating ohmic contact. The ohmic contact here
plays the role of the so-called ‘voltage probe’ introduced
bytheoriststoaccountfordecoherenceandenergyrelax-
p
ation within the scattering approach to quantum trans-
port [15]. Voltage probes act as reservoirs that ab-
FIG. 2. (Color online) (a) Experiment schematic: the outer
sorb all incoming electronic quasiparticles and emit new
EC’srelaxationisforcedbypartlydivertingittowardafloat-
ones with a Fermi statistics at the electrochemical po-
ing ohmic contact, through the intermediate QPC (middle
tential dictated by current conservation. These absorp- split gate) of conductance Gi ∈ [0,1] e2/h. (b) Raw data
tion/emission processes mimic both the quasiparticles’ ∝ ∂fQD(E)/∂E (symbols) at δV = 36 µV are plotted vs
finite quantumlifetime andtheir relaxationtowardther- Vp ∝ E and shifted vertically for different values of Gi. The
mal equilibrium along an EC. An ‘intermediate’ QPC of dashed line is a Fermi fit (Tfit = 36 mK). Continuous lines
tunable conductance G permits to connect controllably are the weighted sums of the data at Gi = 0 and e2/h, as
i predicted by thescattering theory.
the ohmic contact (see Fig. 1 and setup schematic in
Fig.2(a)). Theimpactofsimilarfloatingohmiccontacts
has previously been investigated on current fluctuations
[16]. More recently, their dephasing properties were ex- Figure 2(b) shows as symbols the raw data
plored using an electronic Mach-Zehnder interferometer (∂IQD/∂Vp)/IQmDax vs Vp obtained for different Gi span-
[17]. Here,weuseanexperimentalsetupthatpermitsus ning from zero to e2/h. At G = 0, we observe a double
i
tocharacterizefullythecorrespondingenergyrelaxation. dip that corresponds to a significant but incomplete en-
This experiment is performed as follows: the injection ergy redistribution (as in [11]). In contrast to this non-
QPC is located L = 2.2 µm upstream of the QD, as equilibrium double dip, the energy derivative of a Fermi
shown in Fig. 1, and biased at δV = V −V = 36 µV. function is a single dip whose width and inverse ampli-
1 2
The resulting non-equilibrium outer EC propagates for tudeareproportionaltothetemperature. Intheopposite
1.4 µm along the edge before it reaches the intermediate limit,G =e2/h,weobserveanarrowsingledipfittedus-
i
QPC. It is there partly transmitted with a probability ing a cold Fermi function at T =36 mK (dashed line).
fit
Gh/e2 ∈[0,1]towardafloatingohmiccontactconnected This shows that ECs emitted by the floating ohmic con-
i
to the bottom right ECs shown in Fig. 1. The energy tact are fully thermalized, and that they are not heated
distributionintheouterECisthenmeasuredattheQD, upalongtheirwaybacktotheintermediateQPC.Atpar-
0.8 µm downstream of the intermediate QPC. tial transmissions 0 < Gh/e2 < 1, the data in Fig. 2(b)
i
3
exhibitamorecomplexshape,whichwenowcompareto (a) L
in
the scattering model’s predictions. L
out
Accordingtothescatteringtheory,theenergydistribu-
tion at the intermediate QPC’s output is the sum of the
QD
distribution functions in the twoincoming ECs weighted
by their respective transmission probabilities [15]. Be-
cause interactions are ignored in the scattering theory,
δV
it can be applied to the present experiment only if the
energy relaxationis negligible along the 0.8 µm pathbe-
tween the intermediate QPC and the QD. We showedin (b)
our previous work on the same sample and in the same
(2.2,2.2) (2.2,30)
(L (µm),L (µm)):
experimental conditions that this is a good approxima- in out (2.2,10) (10,10)
tion [13]. However, the energy relaxation is significant
after a propagation path of 2.2 µm [11]. This is taken
into account by using the data measured at G = 0 as
i
thereferencesignalfortheouterECcomingfromthein-
jectionQPC.The correspondingpredictions arethe sum
ofthedataatG =0ande2/hweighted,respectively,by
i
p
the measured 1−Gh/e2 and Gh/e2 (continuous lines
i i
in Fig. 2(b)). We find a good agreement with the data,
without any fitting parameter. In conclusion, we show
here that the connection through a QPC of a floating
ohmic contact used as a voltage probe provides a mean
todrivecontrollablyanon-equilibriumquantumchannel
backtowardequilibrium, inquantitativeagreementwith
the scattering theory.
Freezing energy exchanges. Energy exchanges re-
p
main importantevenwith a single EC [12] (possibly due
tothepresenceoflowenergyspinexcitationsinthebulk FIG. 3. (Color online) (a) The non-equilibrium outer EC
at ν = 1 [18]). However, we show here that energy propagatespartlyalongaclosedinnerEC’sloop,exceptwhen
L
exchanges can be frozen at νL = 2 by closing the in- Lin =Lout. (b) Raw data ∝∂fQD/∂E (symbols) for various
ner EC on itself along the outer EC’s path. The exper- (Lin (µm),Lout (µm)) are shifted vertically for the different
δV ∈{−36,36,54} µV. The data for (2.2,10) are mostly un-
iment schematic is shown in Fig. 3(a). Contrary to the
changed from those for the short direct path (2.2,2.2). This
previous setup, the ECs transmitted through the inter-
implies that energy exchanges are negligible in the outer EC
mediate QPC do not reach the floating ohmic contact.
along the 8 µm closed inner EC loop. The contrast is stark
Instead, they are redirected toward the QD by applying withthedataforthecorrespondingdirectpath(10,10),which
a negative voltage to a surface metal gate barring the exhibit a broad Fermi dip (continuous lines are fits with hot
way (either the bottom right gate in Fig. 1 or another Fermi functions). The observed energy relaxation toward a
gate further away [14]). For G = e2/h, the outer EC is broaddipforthelarger28µminnerECloopof(2.2,30)shows
i
theabove energy exchanges freezing dependson loop size.
fully transmitted and the inner EC fully reflected. As a
result, the inner EC propagates from injection QPC to
QD on a distance L ≃ 2.2 µm, shorter than the outer
in
EC’s Lout ≃ 10 or 30 µm: the extra outer EC propa- therefore that energy exchanges are negligible. The con-
gation path takes place along a closed inner EC loop of trast is stark compared to (10,10), where the outer EC
perimeterLloop ≃8µmor28µm. ForGi =0and2e2/h, propagates on the same length as for (2.2,10), but with
boththeinnerandouterECscopropagatealongthesame a copropagative inner EC. Indeed, the non-equilibrium
length(Lin =Lout)of2.2and10µm,respectively. Inthe doubledipstructuresapparentfor(2.2,2.2)and(2.2,10)
following,wecharacterizetheexperimentalconfiguration arenowwashedoutfor(10,10),andtheenergydistribu-
by the two propagation lengths (Lin (µm),Lout (µm)). tions have relaxed toward hot Fermi functions (continu-
Figure 3(b) shows as symbols (∂I /∂V )/Imax mea- ous lines in Fig. 3(b) are Fermi fits). Consequently, we
QD p QD
sured vs V for δV = −36, 36 and 54 µV applied to the findthatclosingtheinnerECona8µmloopturnsdown
p
injection QPC. The striking feature is that the data for energy exchanges along the outer EC, from a complete
(2.2,2.2) and those for (2.2,10) are essentially identical, energyredistribution(towardahotelectronequilibrium)
at our experimental resolution. This demonstrates di- to a negligible one! A different behavior arises with the
rectly that the energy distribution in the probed outer larger 28 µm inner EC loop realized for (2.2,30), where
EC remains unchanged along the extra 8 µm path, and f (E) relaxestowarda hotFermi function with a tem-
QD
4
perature higher than that for (10,10). This revival of the second quantum Hall channel at ν = 2 [25], clos-
L
energy exchanges shows loop size has an important role. ing the inner EC in loops forbids current fluctuations
We attribute the observed reduction of energy ex- and thereby cancels out this additional contribution. In
changes to the discreteness of energy levels in the closed practice, inner EC closed loops of several microns are
inner EC loop, whose spacing competes with the avail- easily implemented, compatible evenwith opticallithog-
able energy. Indeed, for a 8 µm closed loop and using raphy. From the decrease of the energy relaxation rate
the standard drift velocity v ≈105 m/s at ν =2 [19], bymorethanafactorfour[26],weanticipateanincrease
D L
the energyspacingδE ≈52µeVis largerthan orcom- ofthequantumcoherencelengthfrom20µm[10]tonear
in
parable to e|δV|. Consequently, the available energy in macroscopic length scales, above 80 µm.
the outer EC is not sufficient to excite the discrete inner The authors gratefully acknowledge discussions with
EC’senergylevels[20]. Asaresult,theenergyexchanges A.Anthore,P.Degiovanni,P.Joyez,F.Portier,P.Roche.
between ECs that were previously established at ν =2
L
[11]arehereturnedoff. Thisanalysisisconfirmedbythe
revival of energy exchanges for a 28 µm loop: the corre-
sponding spacing δE ≈ 15 µeV is smaller than e|δV|,
in
therefore there is enough energy available in the outer ∗ Corresponding author: [email protected]
EC to excite the discrete inner EC’s energy levels. [1] See e.g. C. H. Bennett and D. P. DiVincenzo, Nature
404, 247 (2000).
The observation of frozen energy exchanges also pro-
[2] B. I.Halperin, Phys.Rev.B 25, 2185 (1982).
vides new information regarding the inelastic mecha- [3] M. Bu¨ttiker, Phys. Rev.B 38, 9375 (1988).
nisms along an EC at filling factor νL = 2. First, this [4] Y. Ji et al.,Nature422, 415 (2003).
shows that on a 8 µm length scale at the probed en- [5] P. Samuelsson, E. V. Sukhorukov, and M. Bu¨ttiker,
ergies, interactions between electrons within the outer Phys. Rev.Lett. 92, 026805 (2004); S. Ol’khovskaya et
EC are incontrovertibly negligible. The same conclusion al., ibid.101, 166802 (2008).
applies to many other plausible energy exchange mech- [6] I. Nederet al.,Nature448, 333 (2007).
[7] A. Bertoni et al., Phys. Rev.Lett. 84, 5912 (2000);
anisms, including the coupling with the nearby surface
T. M. Stace, C. H. W. Barnes, and G. J. Milburn, ibid.
metalgates,the nuclearspins, the disorderinduced bulk 93, 126804 (2004); R. Ionicioiu, G. Amaratunga, and
statesenclosedintheloop,thenearbycounterpropagat- F. Udrea,Int.J. Mod. Phys. B 15, 125 (2001).
ing ECs and the additional modes within the outer EC [8] L. V. Litvin et al., Phys.Rev.B 75, 033315 (2007).
[21] that are predicted in presence of edge reconstruc- [9] E. Bieri et al.,Phys.Rev. B 79, 245324 (2009).
tionforrealisticsmoothconfinementpotentials[22]. Sec- [10] P. Roulleau et al.,Phys.Rev.Lett. 100, 126802 (2008).
[11] H. le Sueuret al.,Phys. Rev.Lett. 105, 056803 (2010).
ond, this sheds light on the extra energy leak observed
[12] G.Granger, J.P.Eisenstein,andJ.L.Reno,Phys.Rev.
in(10,10)and(30,30)[11]comparedto expectationsfor
Lett. 102, 086803 (2009).
two interacting ECs [23]. This leak strongly suggests [13] C. Altimiras et al.,NaturePhys. 6, 34 (2010).
significant energy transfers toward extra degrees of free- [14] See online supplementary material.
dom. Followingourpreviousworks[11,13],weextracted [15] M. Bu¨ttiker, Phys. Rev.B 33, 3020 (1986).
from f (E) the increase of energy with δV. We find [16] S.Oberholzeret al.,Phys.Rev.Lett.96,046804(2006).
QD
that the electronic energy is identical at our experimen- [17] P. Roulleau et al.,Phys.Rev.Lett. 102, 236802 (2009).
[18] P.Plochockaetal.,Phys. Rev.Lett.102,126806(2009).
talaccuracyinthe configurations(2.2,2.2),(2.2,10)and
[19] V. I. Talyanski et al., J. Phys. Condens. Matter 5, 7643
(2.2,30). This shows that the invoked extra degrees of
(1993).
freedomarefrozenbyclosingthe inner ECintoaloopof
[20] Similar conclusions can be reached beyond a first order
upto30µm[24]. Consequently,thepredictedadditional perturbative treatment of interactions (P. Degiovanni,
modes within the outer EC can be ruled out (although private communication).
not those within the inner EC). The temperatures T , [21] I. L. Aleiner and L. I. Glazman, Phys. Rev.Lett. 72,
exc
corresponding to the energy increase of the probed non- 2935 (1994); C. de C. Chamon and X. G. Wen,
Phys. Rev.B 49, 8227 (1994).
equilibriumexcitations,areshownwitherrorbarsin[14].
[22] D. B. Chklovskii, B. I. Shklovskii, and L. I. Glaz-
We illustrate the above finding with δV = 36 µV, for
man, Phys.Rev.B 46, 4026 (1992); A. H. MacDonald,
which we obtain approximately the same Texc = 89, 91 S. R. E. Yang, and M. D. Johnson, Aust. J. Phys. 46,
and93mKfor,respectively,(2.2,2.2),(2.2,10),(2.2,30), 345 (1993).
whereas we find Texc =73 mK for (10,10). [23] P.Degiovannietal.,Phys. Rev.B81,121302(R)(2010);
Perspectives of the present technique to reduce dras- A. M. Lunde, S. E. Nigg, and M. Bu¨ttiker, ibid. 81,
tically the energy exchange rate include the increase of 041311(R) (2010).
[24] Notethat energyexchangesbetween ECs along theloop
the electronic phase coherence length well beyond the
largest values reported in mesoscopic circuits. Indeed, of (2.2,30) do not change Texc in the stationary regime,
if thereis noenergy leak from the innerEC’s loop.
although the dephasing is known to be driven not only [25] P. Roulleau et al., Phys. Rev. Lett. 101, 186803 (2008);
by energy exchange mechanisms but also by the signif- I. Neder et al., Nature Phys. 3, 534 (2007); I. P. Lev-
icant contribution of the low frequency current noise of kivskyi and E. V. Sukhorukov, Phys.Rev.Lett. 103,
5
036801 (2009). ner EC loop. Consequently, the energy exchange rate is
[26] With two copropagating ECs the energy relaxation is here decreased by more than a factor four.
found significant for a propagation path of 2.2 µm,
whereas it is found negligible along a 8 µm closed in-
Supplementary Material for
‘Tuning Energy Relaxation along Quantum Hall Channels’
C. Altimiras,∗ H. le Sueur,∗ U. Gennser, A. Cavanna, D. Mailly, and F. Pierre†
CNRS, Laboratoire de Photonique et de Nanostructures (LPN) - Phynano team, route de Nozay, 91460 Marcoussis, France
ENERGY DISTRIBUTION SPECTROSCOPY (Lin,Lout) ηG ∆ηG/ηG
1 WITH A QUANTUM DOT (µm) (%)
1 (2.2,2.2) 0.0610 3.5
0 (2.2,10)+ 0.0603 5.0
2 We measuredthe energy distributionfQD in the outer
(2.2,30) 0.0603 5.0
ECusingaQDasatunableenergyfilter[1]. Assuminga
an single active QD level ofenergy Elev, constanttunneling (10,10)+ 0.0606 5.7
J density of states and tunnel rates, the QD current I
QD TABLEI.UsedleverarmsηG andtheirrelativestandarder-
3 in the sequential tunnel regime reads [2]
Srourpsp∆leηmGe/nηtGarfyorFtighuerfere3e)z.inTghoefseynmebrgoyl(e+xc)hpaonignetss(oFuitg.ex3paenrid-
]
l mentalconfigurations,characterizedby(Lin,Lout),forwhich
al IQD =IQmDax(fS(Elev)−fQD(Elev)), (1) thefullQDcalibration wasperformed. Afullcalibration was
h performed only each time a surface gate nearby the QD was
- changedbyanimportantamount. Notethatthedataforen-
s
ergy relaxation with a voltage probe (Fig. 2) were obtained
me where IQmDax is the maximum QD current and fQD (fS) in a different cooldown. The lever arm of this renewed QD
refers to the energy distribution in the outer EC located was found to beηG ≃0.059.
t. at the left (right) side of the QD. fQD and fS can be ex-
a
tractedseparatelybyapplyingasufficientlylargesource-
m
drain voltage, as shown in the right inset of Fig. 1. PRACTICAL REALIZATION OF THE
- DIFFERENT EXPERIMENTAL
d
n The probed energy Elev = E0 − eηGVG was swept CONFIGURATIONS
o using VG, with E0 an unimportant offset and ηG the
c gate voltage-to-energy lever arm calibrated by temper- Theexperimentalconfigurationwastunedin-situwith
[ ature and non-linear QD characterizations. Full details the bias voltages applied to surface metallic gates.

2 regarding the procedure to extract ηG are given in the Supplementary Figure 1 displays an e-beam micro-
v supplementary material of [3]. The precise value of ηG graph of the sample with a lower magnification than in
4 is not of utmost importance in the present work, yet it Fig.1. Itshowsthebottomrightsurfacemetalgatethat
7 entersasascalingfactorforthe overallenergy,including permits to realize the 30 µm propagationpaths between
9
the fit and excess temperatures (e.g. in Supplementary injection QPC and QD.
0
. Figure 3). For completeness, we recapitulate in Table I Supplementary Figure 2 shows how we implement the
7
0 theextractedvaluesofηG forthedifferentconfigurations configuration (Lin =2.2 µm,Lout =10 µm).
with their relative standard errors.
0
1
: As pointed out above, the simple expression of Sup- EXTRACTED EXCESS TEMPERATURES WITH
v plementary Equation 1 assumes only one QD level con- A CLOSED INNER EC LOOP
i
X tributes to IQD, and neglects the energy dependence of
r the electrodes tunneling density ofstatesandofthe tun- Theamountofenergywithintheprobedelectronicex-
a nel rates in and out the QD. In practice, the validity of citations can be extracted from the raw data. This in-
thesehypotheseswerecheckedwithastandardnon-linear formation can be used to shed new lights on the energy
QDcharacterization[2],andbycomparingmixingcham- exchange mechanisms at work.
ber temperatures with fit temperatures obtained within Following our former works [1, 3], we express this en-
this framework (see Figure 2 in [1]). ergy E as an excess temperature defined as:
EC
Importantly, the measured energy distribution is that
T ≡ 6(E −E (δV =0))/νπ2k2, (2)
of 1DCF quasiparticle excitations in the probed edge exc q EC EC B
channel. In particular, the tunnel coupled QD does not
with the ratio E /ν related to the energy distribution
EC
probe the predicted additional edge excitations [4] cor-
f through:
QD
responding to transverse charge oscillations across the
finite width of edge channels [5] (see [1] and references
E /ν = (E−µ)(f (E)−θ(µ−E))dE, (3)
therein for a detailed discussion). EC Z QD
2
(Lin,Lout) δV Tfit
(µm) (µV) (mK)
(10,10) −36 81
(10,10) 36 83
(10,10) 54 129
(2.2,30) −36 101
(2.2,30) 36 108
(2.2,30) 54 145
TABLEII.Fittemperaturescorrespondingtothecontinuous
lines shown in Fig. 3(b).
tial (see supplementary material in [3] for full details on
the procedure to extract T ).
exc
In other words, the excess temperature is the temper-
FIG. 1. E-beam micrograph of the sample. ature difference that gives, at equilibrium, the same in-
crease of energy as that extracted from the measured
non-equilibrium distribution functions.
Supplementary Figure 3 shows as symbols the excess
temperatures from the data with a closed inner EC as
well as in the corresponding configurations with a co-
propagative inner EC (see Fig. 3 for some of the corre-
sponding raw data). For completeness, we also give in
Table II the fit temperatures corresponding to the con-
tinuous lines shown in Fig. 3(b).
The interesting finding here is that the excess temper-
ature is identical, at our experimental accuracy, in the
configurations (2.2,2.2) and (2.2,30) although the lat-
ter shows important energy exchanges. On the contrary,
the configuration (10,10) displays substantially smaller
excess temperatures. This implies that the energy ex-
changes that took place along the extra 28 µm, in pres-
enceofalargeclosedinner ECloop,conservethe energy
FIG. 2. Experimental realization of Lin = 2.2 µm and within the probed outer EC. This was expected in the
Lout = 10 µm. The corresponding schematic is shown in
stationary regime for energy exchanges in the outer EC
Fig. 2(a) and the corresponding data in Fig. 2(b).
mediated by the closed inner EC loop. An important
consequence of this result is that there are negligible en-
ergy leaks toward additional states e.g. in the bulk or
along the edge in this configuration. In particular, the
present analysis permits us to extend to 28 µm propa-
gation lengths the conclusion that the energy exchanges
withthepredictedadditionalmodes[4,6]intheouterEC
arenegligible(seetheparagraphbeforetheconclusionof
the manuscript for 8 µm propagationlengths).
∗ These authors contributed equally to thiswork.
FIG.3. Excesstemperaturesextractedfromtherawdatafor † Corresponding author: [email protected]
various(Lin,Lout)configurations. Fig.3(b)inthemanuscript [1] C. Altimiras, H. le Sueur, U. Gennser, A. Cavanna,
shows some of the corresponding raw data.
D. Mailly, and F. Pierre, NaturePhys. 6, 34 (2010).
[2] L. P. Kouwenhoven, C. M. Marcus, P. L. McEuen,
S. Tarucha, R. M. Westervelt, and N. S. Wingreen, in
with θ(E) the step function, ν the density of states per MesoscopicElectronTransportSeriesE:AppliedSciences,
unitlengthandenergy,andµ theelectrochemicalpoten- Vol. 345, edited by L. L. Sohn, L. P. Kouwenhoven, and
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